Power 5V 1A load using single 3.7V 10Ah li-ion cell, and charge it differently

Can you give me a general idea to have something like this - 1. Power the load from 1 single cell 2. When connected to external power source, like USB, disengage the battery and power the load from USB delivering 5V 1A, and simultaneously charge the cell but at 5A.

But to charge it you would need the correct charge controller for the battery. If you can find one that will run off USB then great. But you will need a way to isolate the battery from the boost converter to charge it. (i.e. you'll need to physically disconnect the battery or have a circuit that detects a USB connection and disconnects the battery via transistor or relay.

As you where told in your previous request for this info. You cannot charge a battery at 5A from a 1A source. If the USB connection is offering 5V @1A then the charge controller will use that 5V to charge the battery. But it will only be able to charge as 1A minus the current being used to run the device. So if the device draws 100mA then the charger will only have about 900mA to work with.

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